You are standing on hard floor. Clothes weigh negligibly. You move only your leg. Everything else remains stationary. Assume you are made of the same material as the average human. Can the force exerted by the one leg alone exceed your body weight?
If you can jump off the ground on one leg then you have successfully exerted a force larger than your weight.
We can see this by just using Newton's second law: $\sum_i F_i=m a$, where $F_i$ is one of the forces, $m$ is the mass of the person, and $a$ is the acceleration of the person
The two forces we are concerned with are the weight, $w$, of the person, which acts downward, and the force $L$ applied by the leg, which acts upward on the person (technically you exert a force $L$ downward on the ground, but by Newton's third law the ground also pushes up on you with force $L$).
I will leave it to you to apply Newton's second law to this system to see how we can get a positive (upward) acceleration.
Yes, you can exert larger force than your weight (if you are strong enough).
The idea that your maximum force on the ground must equate your body weight is a misunderstanding that comes from the intuitive first assumption that forces always balance. Yes, forces balance in many situations, but not when acceleration is involved. If you can make the ground cause an acceleration, then the forces don't balance anymore.
When you stand still, the ground must carry your exact weight. The force you exert on the ground (equal to the reaction normal force $n$ the ground exerts on you) is the same as your weight $w$, described by Newton's 1st law: $$\sum F=0\quad\Leftrightarrow \quad n-w=0\quad\Leftrightarrow \quad n=w$$
When land after a jump, the ground must not only carry your weight but also slow you down to zero speed. It must also cause an acceleration. Or in other words, it must also absorb the momentum you impact with. This is described by Newton's 2nd law: $$\sum F=ma\quad\Leftrightarrow \quad n-w=ma\quad\Leftrightarrow \quad n=w+ma$$ The force you exert now (still equal to the normal force $n$) is larger than your weight $w$.
So, yes, it is definitely possible to exert a larger downwards force on the ground than your weight. Just jump, and you have done so. If you do it in some other way - such as one-leg-stomping in your example - you may find it difficult to impact with high enough momentum that must be absorbed. Then it depends on how strong you are (on how high a speed, your muscles can bring your leg to at impact). But your body weight is no limitation to this.